Abstract

We investigate the existence of a fixed point of certain contractive multivalued mappings of integral type by using the admissible mapping. Our results generalize the several results on the topic in the literature involving Branciari, and Feng and Liu. We also construct some examples to illustrate our results.

1. Preliminaries and Introduction

Fixed point theory is one of the most celebrated research areas that has an application potential not only in nonlinear but also in several branches of mathematics. As a consequence of this fact, several fixed point results have been reported. It is not easy to know, manage, and use all results of this reich theory to get an application. To overcome such problems and clarify the literature, several authors have suggested a more general construction in a way that a number of existing results turn into a consequence of the proposed one. One of the examples of this trend is the investigations of fixed point of certain operator by using the -admissible mapping introduced Samet et al. [1]. This paper has been appreciated by several authors and this trend has been supported by reporting several interesting results; see for example [2–12].

In this paper, we define ()-contractive multivalued mappings of integral type and discuss the existence of a fixed point of such mappings. Our construction and hence results improve, extend, and generalize several results including Branciari [13] and Feng and Liu [14].

In what follows, we recall some basic definitions, notions, notations, and fundamental results for the sake of completeness. Let be a family of nondecreasing functions, such that for each , where is the th iterate of . It is known that, for each , we have for all and for[1]. We denote bythe set of all Lebesgue integrable mappings,which is summable on each compact subset ofand, for each.

Letbe a metric space. We denote bythe space of all nonempty subsets of, bythe space of all nonempty bounded subsets of, and bythe space of all nonempty closed subsets of. Forand,
For every,
We denotebywhen. If, for, there exists a sequenceinsuch that, thenis said to be an orbit ofat. A mapping isorbitally lower semicontinuous at, ifis a sequence inandimplies. Branciari [13] extended the Banach contraction principle [15] in the following way.

Theorem 1. Letbe a complete metric space and letbe a mapping such that
for each, whereand. Thenhas a unique fixed point.

Since then many authors used integral type contractive conditions to prove fixed point theorems in different settings; see for example [12, 16–22]. Feng and Liu [14] extended the result of Branciari [13] to multivalued mappings as follows.

Theorem 2 (see [14]). Letbe a complete metric space and letbe a mapping. Assume that for eachand, there existssuch that
whereand. Thenhas a fixed point inprovidedis lower semicontinuous, with.

Definition 6 (see [2]). Letbe a metric space and letbe a mapping. We say thatis a generalized-contractive if there existssuch that
for eachand, where.

Theorem 7 (see [2]). Letbe a complete metric space and letbe a mapping such that for eachand, we have
where. Assume that there existandsuch that. Moreoveris an-admissible mapping. Then there exists an orbitofatandsuch that. Moreover,if and only ifis lower semicontinuous at.

2. Main Results

In this section, we state and proof our main results. We first give the definition of the following notion.

Definition 8. Letbe a metric space. We say thatis an integral type-contractive mapping if there exist two functionsandsuch that for eachand, there existssatisfying
where.

Example 9. Let be endowed with the usual metric. Defineby
andby
Takeandfor all. Then, for eachand, there existssuch that
Henceis an integral type-contractive mapping. Note that (4) does not hold at.

Definition 10. We say thatis an integral subadditive if, for each, we have

We denote by the class of all integral subadditive functions.

Example 11. Letfor all, for all, andfor all. Then, where.

Definition 12. Letbe a metric space. We say thatis a subintegral type-contractive if there exist two functionsandsuch that for eachand, there existssatisfying
where.

Theorem 14. Letbe a complete metric space and letbe an-admissible subintegral type-contractive mapping. Assume that there existandsuch that. Then there exists an orbitofatandsuch that. Moreover,is a fixed point ofif and only ifisorbitally lower semicontinuous at.

Proof. By the hypothesis, there existandsuch that. Sinceis-admissible, then. Forand, there existssuch that
Sinceis nondecreasing, we have
Asby-admissibility of, we have. Forand, there existssuch that
Sinceis nondecreasing, we have
By continuing the same process, we get a sequenceinsuch that,, and
Lettingin above inequality, we have
Also, we have
which implies that
For any, we have
Since, it can be shown by induction that
From (21) and (26), we have
Sinceit follows thatis Cauchy sequence in. Asis complete, there existssuch thatas. Supposeisorbitally lower semicontinuous at; then
By closedness ofit follows that. Conversely, suppose thatis a fixed point ofthen.

Example 15. Let be endowed with the usual metric. Defineby
andby
Takeandfor all. Then, for eachand, there existssuch that
Henceis a subintegral type-contractive mapping. Clearly,is-admissible. Also, we haveandsuch that. Therefore, all the conditions of Theorem 14 are satisfied andhas infinitely many fixed points. Note that Theorem 2 in Section 1 is not applicable here. For example, takeand.

Definition 16. Letbe a metric space. We say thatis an integral type-contractive mapping if there exist two functionsandsuch that
for eachand, where.

Definition 17. Letbe a metric space. We say thatis a subintegral type-contractive mapping if there exist two functionsandsuch that
for eachand, where.

Proof. By the hypothesis, there existandsuch that. Sinceis-admissible, then. Forand, we have
Since, then we havesuch that
Sinceis nondecreasing, we have
Asby-admissibility of, we have. Thus, we havesuch that
Sinceis nondecreasing, we have
By continuing the same process, we get a sequenceinsuch that,, and
Lettingin above inequality, we have
which implies that
For any, we have
Since, it can be shown by induction that
From (39) and (43), we have
Sinceit follows thatis Cauchy sequence in. Asis complete, there existssuch thatas. Supposeisorbitally lower semicontinuous at; then
Hence,becauseimplies. Conversely, suppose that. Then.

Remark 22. Note that subadditivity of the integral was needed in the proofs of Theorems 14 and 18 in order to obtain inequalities (26) and (43). It is natural to ask wether the conclusions of Theorems 14 and 18 are valid if we replace subintegral contractive conditions (13) and (33) by integral contractive conditions (8) and (32), respectively. Looking at our proofs, we can say that it will be true if the inequalities (26) and (43) hold. Here we would like to mention that many authors (see for example [14, 23]) while proving the results on integral contractions have not assumed that the integral is subadditive but indeed they used the subadditivity of the integral in the proofs of their results while obtaining the inequalities comparable to inequalities (26) and (43).

3. Application

In this section, we obtain some fixed point results for partially ordered metric spaces, as consequences of aforementioned results. Moreover, we apply our result to prove the existence of solution for an integral equation.

Letandbe subsets of a partially ordered set. We say that, if for eachand, we have.

Theorem 23. Letbe a complete ordered metric space and letbe a mapping such that for eachandwith, there existssatisfying
whereand. Assume that there existandsuch that. Also, assume thatimplies. Then there exists an orbitofatandsuch that. Moreover,is a fixed point ofif and only ifisorbitally lower semicontinuous at.

Proof. Defineby
By using hypothesis of corollary and definition of, we have. Asimplies, by using the definitions ofand, we have thatimplies. Moreover, it is easy to check thatis an integral type-contractive mapping. Therefore, by Theorem 14, there exists an orbitofatandsuch that. Moreover,is a fixed point ofif and only ifisorbitally lower semicontinuous at.

Corollary 24. Letbe a complete ordered metric space and letbe a nondecreasing mapping such that, for eachwith, we have
where. Assume that there existssuch that. Then there exists an orbitofatandsuch that. Moreover,is a fixed point ofif and only ifisorbitally lower semicontinuous at.

Consider an integral equation of the form
where is continuous and nondecreasing.

Theorem 25. Assume that(i)for, withfor each, we have
for each, where;(ii)for each, there exists such that
Then there exists an iterative sequence, starting from, and such that. Moreover,is a solution of (53) if and only ifis lower semicontinuous at, where.

Proof. It is easy to see that is complete with respect to the metric. We define partial ordering onas followsif and only iffor each. Defineby, where, for each. From (ii), we have. For, letand; that is, and, for each. Then, for eachwith, we have
That is, for eachwith. Clearly,is nondecreasing. Therefore, all conditions of Corollary 24 hold and the conclusions follow from Corollary 24.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgment

The authors are grateful to the reviewers for their careful readings and useful comments.